“If … there were no solid bodies in nature there would be no geometry.”-Poincaré

A while ago, I discussed the mystery of why matter should be the source of gravity. To date, this remains simply an empirical fact. The deep insight of general relativity – that gravity is the geometry of space and time – only provides us with a modern twist: why should matter dictate the geometry of space-time?

There is a possible answer, but it requires us to understand space-time in a different way: as an abstraction that is derived from the properties of matter itself. Under this interpretation, it is perfectly natural that matter should affect space-time geometry, because space-time is not simply a stage against which matter dances, but is fundamentally dependent on matter for its existence. I will elaborate on this idea and explain how it leads to a new avenue of approach to quantum gravity.

First consider what we mean when we talk about space and time. We can judge how far away a train is by listening to the tracks, or gauge how deep a well is by dropping a stone in and waiting to hear the echo. We can tell a mountain is far away just by looking at it, and that the cat is nearby by tripping over it. In all these examples, an interaction is necessary between myself and the object, sometimes through an intermediary (the light reflected off the mountain into my eyes) and sometimes not (tripping over the cat). Things can also be far away in time. I obviously cannot interact with people who lived in the past (unless I have a time machine), or people who have yet to be born, even if they stood (or will stand) exactly where I am standing now. I cannot easily talk to my father when he was my age, but I can almost do it, just by talking to him now and asking him to remember his past self. When we say that something is far away in either space or time, what we really mean is that it is hard to interact with, and this difficulty of interaction has certain universal qualities that we give the names `distance’ and `time’.
It is worth mentioning here, as an aside, that in a certain sense, the properties of `time’ can be reduced to properties of `distance’ alone. Consider, for instance, that most of our interactions can be reduced to measurements of distances of things from us, at a given time. To know the time, I invariably look at the distance the minute hand has traversed along its cycle on the face of my watch. Our clocks are just systems with `internal’ distances, and it is the varying correspondence of these `clock distances’ with the distances of other things that we call the `time’. Indeed, Julian Barbour has developed this idea into a whole research program in which dynamics is fundamentally spatial, called Shape Dynamics.

Sigmund Freud Museum, Wien – Peter Kogler

So, if distance and time is just a way of describing certain properties of matter, what is the thing we call space-time?

We now arrive at a crucial point that has been stressed by philosopher Harvey Brown: the rigid rods and clocks with which we claim to measure space-time do not really measure it, in the traditional sense of the word `measure’. A measurement implies an interaction, and to measure space-time would be to grant space-time the same status as a physical body that can be interacted with. (To be sure, this is exactly how many people do wish to interpret space-time; see for instance space-time substantivalism and ontological structural realism).

Brown writes:“One of Bell’s professed aims in his 1976 paper on `How to teach relativity’ was to fend off `premature philosophizing about space and time’. He hoped to achieve this by demonstrating with an appropriate model that a moving rod contracts, and a moving clock dilates, because of how it is made up and not because of the nature of its spatio-temporal environment. Bell was surely right. Indeed, if it is the structure of the background spacetime that accounts for the phenomenon, by what mechanism is the rod or clock informed as to what this structure is? How does this material object get to know which type of space-time — Galilean or Minkowskian, say — it is immersed in?” [1]

I claim that rods and clocks do not measure space-time, they embody space-time. Space-time is an idealized description of how material rods and clocks interact with other matter. This distinction is important because it has implications for quantum gravity. If we adopt the more popular view that space-time is an independently existing ontological construct, it stands to reason that, like other classical fields, we should attempt to directly quantise the space-time field. This is the approach adopted in Loop Quantum Gravity and extolled by Rovelli:

“Physical reality is now described as a complex interacting ensemble of entities (fields), the location of which is only meaningful with respect to one another. The relation among dynamical entities of being contiguous … is the foundation of the space-time structure. Among these various entities, there is one, the gravitational field, which interacts with every other one and thus determines the relative motion of the individual components of every object we want to use as rod or clock. Because of that, it admits a metrical interpretation.” [2]

One of the advantages of this point of view is that it dissolves some seemingly paradoxical features of general relativity, such as the fact that geometry can exist without (non-gravitational) matter, or the fact that geometry can carry energy and momentum. Since gravity is a field in its own right, it doesn’t depend on the other fields for its existence, nor is there any problem with it being able to carry energy. On the other hand, this point of view tempts us into framing quantum gravity as the mathematical problem of quantising the gravitational field. This, I think, is misguided.

I propose instead to return to a more Machian viewpoint, according to which space-time is contingent on (and not independent of) the existence of matter. Now the description of quantum space-time should follow, in principle, from an appropriate description of quantum matter, i.e. of quantum rods and clocks. From this perspective, the challenge of quantum gravity is to rebuild space-time from the ground up — to carry out Einstein’s revolution a second time over, but using quantum material as the building blocks.

Ernst Mach vs. Max Ernst. Get it right, folks.

My view about space-time can be seen as a kind of `pulling oneself up by one’s bootstraps’, or a Wittgenstein’s ladder (in which one climbs to the top of a ladder and then throws the ladder away). It works like this:Step 1: define the properties of space-time according to the behaviour of rods and clocks.Step 2: look for universal patterns or symmetries among these rods and clocks.Step 3: take the ideal form of this symmetry and promote it to an independently existing object called `space-time’.Step 4: Having liberated space-time from the material objects from which it was conceived, use it as the independent standard against which to compare rods and clocks.

Seen in this light, the idea of judging a rod or a clock by its ability to measure space or time is a convenient illusion: in fact we are testing real rods and clocks against what is essentially an embodiment of their own Platonic ideals, which are in turn conceived as the forms which give the laws of physics their most elegant expression. A pertinent example, much used by Julian Barbour, is Ephemeris time and the notion of a `good clock’. First, by using material bodies like pendulums and planets to serve as clocks, we find that the motions of material bodies approximately conform to Newton’s laws of mechanics and gravitation. We then make a metaphysical leap and declare the laws to be exactly true, and the inaccuracies to be due to imperfections in the clocks used to collect the data. This leads to the definition of the `Ephemeris time’, the time relative to which the planetary motions conform most closely to Newton’s laws, and a `good clock’ is then defined to be a clock whose time is closest to Ephemeris time.

The same thing happens in making the leap to special relativity. Einstein observed that, in light of Maxwell’s theory of electromagnetism, the empirical law of the relativity of motion seemed to have only a limited validity in nature. That is, assuming no changes to the behaviour of rods and clocks used to make measurements, it would not be possible to establish the law of the relativity of motion for electrodynamic bodies. Einstein made a metaphysical leap: he decided to upgrade this law to the universal Principle of Relativity, and to interpret its apparent inapplicability to electromagnetism as the failure of the rods and clocks used to test its validity. By constructing new rods and clocks that incorporated electromagnetism in the form of hypothetical light beams bouncing between mirrors, Einstein rebuilt space-time so as to give the laws of physics a more elegant form, in which the Relativity Principle is valid in the same regime as Maxwell’s equations.

Ladder for Booker T. Washington – Martin Puryear

By now, you can guess how I will interpret the step to general relativity. Empirical observations seem to suggest a (local) equivalence between a uniformly accelerated lab and a stationary lab in a gravitational field. However, as long as we consider `ideal’ clocks to conform to flat Minkowski space-time, we have to regard the time-dilated clocks of a gravitationally affected observer as being faulty. The empirical fact that observers stationary in a gravitational field cannot distinguish themselves (locally) from uniformly accelerated observers then seems accidental; there appears no reason why an observer could not locally detect the presence of gravity by comparing his normal clock to an `ideal clock’ that is somehow protected from gravity. On the other hand, if we raise this empirical indistinguishability to a matter of principle – the Einstein Equivalence Principle – we must conclude that time dilation should be incorporated into the very definition of an `ideal’ clock, and similarly with the gravitational effects on rods. Once the ideal rods and clocks are updated to include gravitational effects as part of their constitution (and not an interfering external force) they give rise to a geometry that is curved. Most magically of all, if we choose the simplest way to couple this geometry to matter (the Einstein Field Equations), we find that there is no need for a gravitational force at all: bodies follow the paths dictated by gravity simply because these are now the inertial paths followed by freely moving bodies in the curved space-time. Thus, gravity can be entirely replaced by geometry of space-time.

As we can see from the above examples, each revolution in our idea of space-time was achieved by reconsidering the nature of rods and clocks, so as to make the laws of physics take a more elegant form by incorporating some new physical principle (eg. the Relativity and Equivalence principles). What is remarkable is that this method does not require us to go all the way back to the fundamental properties of matter, prior to space-time, and derive everything again from scratch (the constructive theory approach). Instead, we can start from a previously existing conception of space-time and then upgrade it by modifying its primary elements (rods and clocks) to incorporate some new principle as part of physical law (the principle theory approach). The question is, will quantum gravity let us get away with the same trick?

I’m betting that it will. The challenge is to identify the empirical principle (or principles) that embody quantum mechanics, and upgrade them to universal principles by incorporating them into the very conception of the rods and clocks out of which general relativistic space-time is made. The result will be, hopefully, a picture of quantum geometry that retains a clear operational interpretation. Perhaps even Percy Bridgman, who dismissed the Planck length as being of “no significance whatever” [3] due to its empirical inaccessibility, would approve.

I recently returned to my roots, contributing to a new paper with Tim Ralph (who was my PhD advisor) on the very same topic that formed a major part of my PhD. Out of laziness, let me dig up the relevant information from an earlier post:

“The idea for my PhD thesis comes from a paper that I stumbled across as an undergraduate at the University of Melbourne. That paper, by Tim Ralph, Gerard Milburn and Tony Downes of the University of Queensland, proposed that Earth’s own gravitational field might be strong enough to cause quantum gravity effects in experiments done on satellites. In particular, the difference between the strength of gravity at ground-level and at the height of the orbiting satellite might be just enough to make the quantum particles on the satellite behave in a very funny non-linear way, never before seen at ground level. Why might this happen? This is where the story gets bizarre: the authors got their idea after looking at a theory of time-travel, proposed in 1991 by David Deutsch. According to Deutsch’s theory, if space and time were bent enough by gravity to create a closed loop in time (aka a time machine), then any quantum particle that travelled backwards in time ought to have a very peculiar non-linear behaviour. Tim Ralph and co-authors said: what if there was only a little bit of space-time curvature? Wouldn’t you still expect just a little bit of non-linear behaviour? And we can look for that in the curvature produced by the Earth, without even needing to build a time-machine!”

In our recent paper in New Journal of Physics, for the special Focus on Gravitational Quantum Mechanics, Tim and I re-examined the `event formalism’ (the fancy name for the nonlinear model in question) and we derived some more practical numerical predictions and ironed out a couple of theoretical wrinkles, making it more presentable as an experimental proposal. Now that there is growing interest in quantum gravity phenomenology — that is, testable toy models of quantum gravity effects — Tim’s little theory has an excitingly real chance of being tested and proven either right or wrong. Either way, I’d be curious to know how it turns out! On one hand, if quantum entanglement survives the test, the experiment would stand as one of the first real confirmations of quantum field theory in curved space-time. On the other hand, if the entanglement is destroyed by Earth’s gravitational field, it would signify a serious problem with the standard theory and might even confirm our alternative model. That would be great too, but also somewhat disturbing, since non-linear effects are known to have strange and confusing properties, such as violating the fabled uncertainty principle of quantum mechanics.

You can see my video debut here, in which I give an overview of the paper, complete with hand-drawn sketches!

(Actually there is a funny story attached to the video abstract. The day I filmed the video for this, I had received a letter informing me that my application for renewal of my residence permit in Austria was not yet complete — but the permit itself had expired the previous day! As a result, during the filming I was half panicking at the thought of being deported from the country. In the end it turned out not to be a problem, but if I seem a little tense in the video, well, now you know why.)

This is one of those questions that has always bugged me.
Suppose that, somewhere in the universe, there is a very large closed box made out of some kind of heavy, neutral matter. Inside this box a civilisation of intelligent creatures have evolved. They are made out of normal matter like you and me, except that for some reason they are very light — their bodies do not contain much matter at all. What’s more, there are no other heavy bodies or planets inside this large box aside from the population of aliens, whose total mass is too small to have any noticeable effect on the gravitational field. Thus, the only gravitational field that the aliens are aware of is the field created by the box itself (I’m assuming there are no other massive bodies near to the box).

Setting aside the obvious questions about how these aliens came to exist without an energy source like the sun, and where the heck the giant box came from, I want to examine the following question: in principle, is there any way that these aliens could figure out that matter is the source of gravitational fields?

Now, to make it interesting, let us assume the density of the box is not uniform, so there are some parts of its walls that have a stronger gravitational pull than others. Our aliens can walk around on these parts of the walls, and in some parts the aliens even become too heavy to support their own weight and get stuck until someone rescues them. Elsewhere, the walls of the box are low density and so the gravitational attraction to them is very weak. Here, the aliens can easily jump off and float away from the wall. Indeed, the aliens spend much of their time floating freely near the center of the box where the gravitational fields are weak. Apart from that, the composition of the box itself does not change with time and the box is not rotating, so the aliens are quickly able to map out the constant gravitational field that surrounds them inside the box, with its strong and weak points.

Like us, the aliens have developed technology to manipulate the electromagnetic field, and they know that it is the electromagnetic forces that keeps their bodies intact and stops matter from passing through itself. More importantly, they can accelerate objects of different masses by pushing on them, or applying an electric force to charged test bodies, so they quickly discover that matter has inertia, measured by its mass. In this way, they are able to discover Newton’s laws of mechanics. In addition, their experiments with electromagnetism and light eventually lead them to upgrade their picture of space-time, and their Newtonian mechanics is replaced by special relativistic mechanics and Maxwell’s equations for the electromagnetic field.

So far, so good! Except that, because they do not observe any orbiting planets or moving gravitating bodies (their own bodies being too light to produce any noticible attractive forces), they still have not reproduced Newtonian gravity. They know that there is a static field permeating space-time, called the gravitational field, that seems to be fixed to the frame of the box — but they have no reason to think that this gravitational force originates from matter. Indeed, there are two philosophical schools of thought on this. The first group holds that the gravitational field is to be thought of analogously to the electromagnetic field, and is therefore sourced by special “gravitational charges”. It was originally claimed that the material of the box itself carries gravitational charge, but scrapings of the material from the box revealed it to be the same kind of matter from which the aliens themselves were composed (let’s say Carbon) and the scrapings themselves seemed not to produce any gravitational fields, even when collected together in large amounts of several kilograms (a truly humungous weight to the minds of the aliens, whose entire population combined would only weigh ten kilograms). Some aliens pointed out that the gravitational charge of Carbon might be extremely weak, and since the mass of the entire box was likely to be many orders of magnitude larger than anything they had experienced before, it is possible that its cumulative charge would be enough to produce the field. However, these aliens were criticised for making ad-hoc modifications to their theory to avoid its obvious refutation by the kilograms-of-Carbon experiments. If gravity is analogous to the electromagnetic force — they were asked with a sneer — then why should it be so much weaker than electromagnetism? It seemed rather too convenient.

Some people suggested that the true gravitational charge was not Carbon, but some other material that coated the outside of the box. However, these people were derided even more severely than were the Carbon Gravitists (as they had become known). Instead, the popular scientific consensus shifted to a modern idea in which the gravitational force was considered to be a special kind of force field that simply had no source charges. It was a God-given field whose origin and patterns were not to be questioned but simply accepted, much like the very existence of the Great Box itself. This following gained great support when someone made a great discovery: the gravitational force could be regarded as the very geometry of spacetime itself.

The motivation for this was the peculiar observation, long known but never explained, that massive bodies always had the same acceleration in the gravitational field regardless of their different masses. A single alien falling towards one of the gravitating walls of the box would keep speed perfectly with a group of a hundred Aliens tied together, despite their clearly different masses. This dealt a crushing blow to the remnants of the Carbon Gravitists, for it implied that the gravitational charge of matter was exactly proportional to its inertial mass. This coincidence had no precedent in electromagnetism, where it was known that bodies of the same mass could have very different electric charges.

Under the new school of thought, the gravitational force was reinterpreted as the background geometry of space-time inside the box, which specified the inertial trajectories of all massive bodies. Hence, the gravitational force was not a force at all, so it was meaningless to ascribe a “gravitational charge” to matter. Tensor calculus was developed as a natural extension of special relativity, and the aliens derived the geodesic equation describing the motion of matter in a fixed curved space-time metric. The metric of the box was mapped out with high precision, and all questions about the universe seemed to have been settled.

Well, almost all. Some troublesome philosophers continued to insist that there should be some kind of connection between space-time geometry and matter. They wanted more than just the well-known description of how geometry caused matter to move: they tried to argue that matter should also tell space-time how to curve.

“Our entire population combined only weighs a fraction of the mass of the box. What would happen if there were more matter available to us? What if we did the Carbon-kilogram experiment again, but with 100 kilograms? Or a million? Surely the presence of such a large amount of matter would have an effect on space-time itself?”

But these philosophers were just laughed at. Why should any amount of matter affect the eternal and never-changing space-time geometry? Even if the Great Box itself were removed, the prevailing thought was that the gravitational field would remain, fixed as it was in space-time and not to any material source. So they all lived happily ever after, in blissful ignorance of the gravitational constant G, planetary orbits, and other such fantasies.

***

Did you find this fairytale disturbing? I did. It illustrates what I think is an under-appreciated uncomfortable feature of our best theories of gravity: they all take the fact that matter generates gravity as a premise, without justification apart from empirical observation. There’s nothing strictly wrong with this — we do essentially the same thing in special relativity when we take the speed of light to be constant regardless of the motion of its source, historically an empirically determined fact (and one that was found quite surprising).

However, there is a slight difference: one can in principle argue that the speed of light should be reference-frame independent from philosophical grounds, without appealing to empirical observations. Roughly, the relativity principle states that the laws of physics should be the same in all frames of motion, and from among the laws of physics we can include the non-relativistic equations of the electromagnetic field, from which the constant speed of light can be derived from the electric and magnetic constants of the vacuum. As far as I know, there is no similar philosophical grounding for the connection between matter and geometry as embodied by the gravitational constant, and hence no compelling reason for our hypothetical aliens to ever believe that matter is the source of space-time geometry.

Could it be that there is an essential piece missing from our accounts of the connection between matter and space-time? Or are our aliens are doomed by their unfortunately contrived situation, never to deduce the complete laws of the universe?

One of the best things about the internet is that it gives the public unprecedented access to the cutting edge of science. For example, on Friday I uploaded my entire PhD thesis, “Causality Violation and Nonlinear Quantum Mechanics”, to the arXiv, where it can now be read by my contemporaries, colleagues, parents, neighbours and any pets that can read human-speak. The downside is, as my parents pointed out, my thesis is not written in English but in gobbledegook that makes no sense to anybody outside of the quantum physics community. So, while most work at the cutting edge of physics is available to anyone to read, almost nobody can understand it — and those who can, have access to journal subscriptions anyway! Oh, the irony. In order to remedy this problem, I’ve decided to translate my thesis from jargon into words that most people can understand (I’m not the first one to have this idea – check out Scott Aaronson’s description of his own research using the 1000 most common words in English).

Like any PhD thesis, my work is on a really small, pigeonhole topic that is part of a much bigger picture — like one brick in the wall of a huge building that is being constructed. I’ll start by talking about the whole building, which represents all of physics, and then slowly zoom in on the single, tiny brick that I focused on intensely for three and a half years. One thing about doing a PhD is that it teaches you to be very Zen about your work. We can’t all expect to be like Einstein, who single-handedly constructed an entire wing of the building — us regular scientists have to content ourselves with the laying of a single brick. In the end, the whole structure will only stand if every single brick is placed carefully and correctly. Sherlock Holmes once observed that, from a single drop of water, “a logician could infer the possibility of an Atlantic or a Niagara without having seen or heard of one or the other. So all life is a great chain, the nature of which is known whenever we are shown a single link of it.” In the same way, the contribution of just a single brick to the building of physics is significant. After all, one cannot know how to place the individual brick without having first consulted the blueprint of the entire building. In this sense, the individual brick is as significant as the entire structure.

But enough philosophical mumbo-jumbo! Let’s get concrete. Modern physics can be roughly divided up into two separate theories. The first is Einstein’s theory of gravity, called General Relativity. This theory applies to all heavy objects that are affected by gravity, like planets, galaxies, and your mum. The theory says that heavy objects cause space and time to bend and curve around them in a special way. When the objects move through this curved space-time, they follow paths that bring them closer together, creating the appearance of an attractive force — the “force” normally known as gravity. If you want to fly rockets through space or deduce how the galaxy formed, or predict how heavy objects move around each other in space, you need this theory.

The other theory is Quantum Mechanics. This governs the behaviour of things that are very small, like atoms and the smaller parts of atoms. It turns out that even light is composed of a vast number of very small particles, called photons. If you have a very weak source of light that emits only a few photons at a time, it too is described by quantum mechanics. All quantum particles possess wave-like behaviour under certain conditions, which is described by something called the “wave-function” of the particle. Since we are made entirely out of atoms, you might ask: how come our bodies are not governed by quantum mechanics too? The answer is that, although in theory there could be a wave-function for larger objects like human bodies, in practice it is extremely difficult to create the special conditions needed to see the quantum behaviour of such large collections of atoms. The process whereby quantum effects become harder to observe in bigger objects is called “decoherence”, and it is still not completely understood.

Now comes the interesting part: it turns out that gravity is very weak. We can feel the Earth’s gravitational pull because the Earth is really, really heavy. But, in theory, you and I are also heavy enough to cause space-time to bend, just a little bit, around our bodies. This means you should be pulled towards me by gravity, and I towards you. However, because we are not nearly as heavy as the Earth, our gravitational pull on each other is far too weak for us to notice (though people have observed the gravitational attraction between large, heavy balls suspended by wires). By the time you get down to molecules, atoms and quantum mechanics, the gravitational force is so tiny that it can be completely ignored. On the other hand, because of decoherence, quantum mechanics can usually be ignored for anything much larger than molecules. This means that it is very rare to find a situation in which both quantum effects and gravitational effects both need to be taken into account; it is almost always a case of just one or the other.

The trouble is, if there were an in-between situation where both theories become significant, we do not know how quantum mechanics and gravity would overlap. One example is when a star becomes so heavy that it collapses under its own gravity into a black hole. The singularity at the centre of the hole is small enough that quantum mechanics should become important. As for being heavy, well, a black hole is so heavy that even light cannot go fast enough to escape from its gravitational attraction. Another in-between case would occur if we ever manage to make instruments that can measure space and time to extremely high accuracy, at a level called the “Planck scale”. If we are ever able to measure such tiny distances and times, we might expect to observe the curving of space-time due to very low-mass objects like atoms. So far, such precision is beyond our technology, and black holes have only been observed very indirectly. But when we finally do begin to probe these things, we will need a theory to describe them that incorporates both gravity and quantum mechanics together — a theory of “quantum gravity”. But despite roughly a hundred years of effort, we still do not have such a theory.

Why is it so hard to do quantum mechanics and gravity at the same time? This question alone is the subject of much debate. For now, you’ll just have to take my word that you can’t simply mash the two together (it has something to do with the fact that “space-time” is no longer clearly defined at the Planck scale). One approach is to consider a specific example of something that needs quantum gravity to describe it, like a black hole, and then try to develop a kind of “toy” theory of quantum gravity that describes only that particular situation. Once you have enough toy theories for different situations, you might be able to stick them together into a proper theory that includes all of them as special cases. This is easier said than done — it relies on us being able to come up with a wide variety of “thought experiments” that combine different aspects of quantum mechanics and gravity. But thought experiments like these are very rare: you’ve got black holes, Roger Penrose’s idea of a massive object in a quantum superposition, and a smattering of lesser known ideas. Are there any more?

This is where I come in. The idea for my PhD thesis comes from a paper that I stumbled across as an undergraduate at the University of Melbourne. The paper, by Tim Ralph, Gerard Milburn and Tony Downes of the University of Queensland, proposed that Earth’s own gravitational field might be strong enough to cause quantum gravity effects in experiments done on satellites. In particular, the difference between the strength of gravity at ground-level and at the height of the orbiting satellite might be just enough to make the quantum particles on the satellite behave in a very funny “non-linear” way, never before seen at ground level. Why might this happen? This is where the story gets bizarre: the authors got their idea after looking at a theory of time-travel, proposed in 1991 by David Deutsch. According to Deutsch’s theory, if space and time were bent enough by gravity to create a closed loop in time (aka a time machine), then any quantum particle that travelled backwards in time ought to have a very peculiar “non-linear” behaviour. Tim Ralph and co-authors said: what if there was only a little bit of space-time curvature? Wouldn’t you still expect just a little bit of non-linear behaviour? And we can look for that in the curvature produced by the Earth, without even needing to build a time-machine!

I was so fascinated by this idea that I immediately wrote to Tim Ralph. After some discussions, I visited Brisbane for the first time to meet him, and soon afterwards I began my PhD at the Uni of Queensland under Tim’s supervision. My first task was to understand Deutsch’s model of time-travel in more detail. (Unfortunately, the idea of time-travel is notorious for attracting crackpots, so if you want to do legitimate research on the topic you have to couch it in jargon to convince your colleagues that you have not gone crazy — hence the replacement of “time-travel” with the more politically-correct term “causality violation” in the title of my thesis). David Deutsch is best known for being one of the founding fathers of the idea of a quantum computer. That gave him enough credibility amongst physicists that we were willing to listen to his more eccentric ideas on things like Everett’s many-worlds interpretation of quantum mechanics, and the quantum mechanics of time-travel. On the latter topic, Deutsch made the seminal observation that certain well-known paradoxes of time travel — for instance, what happens if you go back in time and kill your past self before he/she enters the time machine — could be fixed by taking into account the wave-function of a quantum particle as it travels back in time.

Source: xkcd.com

Roughly speaking, the closed loop in time causes the wave-function of the particle to loop back on itself, allowing the particle to “interact with itself” in a non-linear way. Just like Schrödinger’s cat can be both dead and alive at the same time in a quantum superposition, it is possible for the quantum particle to “kill it’s past self” and “not kill it’s past self” at the same time, thereby apparently resolving the paradox (although, you might be forgiven for thinking that we just made it worse…).

Here was a prime example of a genuine quantum gravity effect: space-time had to be curved in order to create the time-machine, but quantum mechanics had to be included to resolve the paradoxes! Could we therefore use this model as a new thought experiment for understanding quantum gravity effects? Luckily for me, there were a few problems with Deutsch’s model that still needed to be ironed out before we could really take seriously the experiment proposed by Ralph, Milburn and Downes. First of all, as I mentioned, Deutsch’s model introduced non-linear behaviour of the quantum particle. But many years earlier, Nicolas Gisin and others had argued that any non-linear effects of this type should allow you to send signals faster than light, which seemed to be against the spirit of Einstein’s theory of relativity. Secondly, Deutsch’s model did not take into account all of the quantum properties of the particle, such as the spread of the particle’s wave function in space and time. Without that, it remained unclear whether the non-linear behaviour should persist in Earth’s gravitational field, as Ralph and co-authors had speculated, or whether the space-time spread of the wave-function would some how “smear out” the non-linearity until it disappeared altogether.

In my thesis, I showed that it might be possible to keep the non-linear behaviour of the Deutsch model while also ensuring that no signals could be sent faster than light outside of the time-machine. My arguments were based on some work that had already been done by others in a different context — I was able to adapt their work to the particular case of Deutsch’s model. In addition, I re-formulated Deutsch’s model to describe what happens to pulses of light, such that the space-time spread of the light could be taken into account together with the other quantum properties of light. Using this model, I showed that even if a pulse of light were sent back in time without interacting with its past self at all (no paradoxes), the wave-function of the light would still behave in a non-linear way. Using my model, I was able to describe exactly when the non-linear effects would get “smeared out” by the wave-function, and confirmed that the non-linear effects might still be observable in Earth’s gravitational field without needing a time-machine, thereby lending further support to the speculative work of Ralph and co. that had started it all.

So, that’s my PhD in a nutshell! Where to next? Right now I have decided to calm down a little bit and steer towards less extreme examples of a quantum gravity thought experiments. In particular, rather than looking at outright “causality violation”, I am investigating a peculiar effect called “indefinite causality”, in which space-time is not quite curved enough to send anything backwards in time, but where it is also not clear which events are causes and which ones are their effects. Hopefully, I’ll be able to understand how quantum mechanics fits into this weird picture — but that’s a topic for another post.

This is the first of a series of posts in which I will argue that physicists can gain insight into the puzzles of quantum gravity if we adopt a philosophy I call operationalism. The traditional interpretation of operationalism by philosophers was found to be lacking in several important ways, so the concept will have to be updated to a modern context if we are to make use of it, and its new strengths and limitations will need to be clarified. The goal of this first post is to introduce you to operationalism as it was originally conceived and as I understand it. Later posts will explain the areas in which it failed as a philosophical doctrine, and why it might nevertheless succeed as a tool in theoretical physics, particularly in regard to quantum gravity [1].

Operationalism started with Percy Williams Bridgman. Bridgman was a physicist working in the early 20th century, at the time when the world of physics was being shaken by the twin revolutions of relativity and quantum mechanics. Einstein’s hand was behind both revolutions: first through the publication of his theory of General Relativity in 1916, and second for explaining the photoelectric effect using things called quanta, which earned him the Nobel prize in 1921. This upheaval was a formative time for Bridgman, who was especially struck by Einstein’s clever use of thought experiments to derive special relativity.

Einstein had realized that there was a problem with the concept of `simultaneity’. Until then, everybody had taken it for granted that if two events are simultaneous, then they occur at the same time no matter who is observing them. But Einstein asked the crucial question: how does a person know that two events happened at the same time? To answer it, he had to adopt an operational definition of simultaneity: an observer traveling at constant velocity will consider two equidistant events to be simultaneous if beams of light emitted from each event reach the location of the observer at the same time, as measured by the observer’s clock (this definition can be further generalised to apply to any pair of events as seen by an observer in arbitrary motion).

From this, one can deduce that the relativity principle implies the relativity of simultaneity: two events that are simultaneous for one observer may not be simultaneous for another observer in relative motion. This is one of the key observations of special relativity. Bridgman noticed that Einstein’s deep insight relied upon taking an abstract concept, in this case simultaneity, and grounding it in the physical world by asking `what sort of operations must be carried out in order to measure this thing’?

For his own part, Bridgman was a brilliant experimentalist who won the Nobel prize in 1946 for his pioneering work on creating extremely high pressures in his laboratory. Using state-of-the-art technology, he created pressures up to 100,000 atmospheres, nearly 100 times greater than anybody before him, and then did what any good scientist would do: he put various things into his pressure chamber to record what happened to them. Mostly, as you might expect, they got squished. At pressures beyond 25,000 atmospheres, steel can be molded like play-dough; at 50,000 atmospheres all normal liquids have frozen solid. (Of course, Bridgman’s vessel had to be very small to withstand such pressure, which limited the things he could put in it). But Bridgman faced a unique problem: the pressures that he created were so high that he couldn’t use any standard pressure gauge to measure the pressures in his lab because the gauge would basically get squished like everything else. The situation is the same as trying to measure the temperature of the sun using a regular thermometer: it would explode and vaporize before you could even take a proper reading. Consequently, Bridgman had no scientific way to tell between `really high pressure’ and `really freaking high pressure’, so he was forced to design completely new ways of measuring pressure in his laboratory, such as looking at the phase transition of the element Bismuth and the resistivity of the alloy Manganin [2]. This led him to wonder: what does a concept like `pressure’ or`temperature’really mean in the absence of a measuring technique?

Bridgman proposed that quantities measured by different operations should always be regarded as being fundamentally different, even though they may coincide in certain situations. This led to a minor problem in the definitions of quantities. The temperature of a cup of water is measured by sticking a thermometer in it. The temperature of the sun is measured by looking at the spectrum of radiation emitted from it. If these quantities are measured by such different methods in different regimes, why do we call them both `temperature’? In what sense are our operations measuring the same thing? The solution, according to Bridgman, is that there is a regime in between the two in which both methods of measuring temperature are valid – and in this regime the two measurements must agree. The temperature of molten gold could potentially be measured by the right kind of thermometer, as well as by looking at its radiation spectrum, and both of these methods will give the same temperature. This allows us to connect the concept of temperature on the sun to temperature in your kitchen and call them by the same name.

This method of `patching together’ different ways of measuring the same quantity is reminiscent of placing co-ordinate patches on manifolds in mathematical physics. In general, there is no way to cover an entire manifold (representing space-time for example) with a single set of co-ordinates that are valid everywhere. But we can cover different parts of the manifold in patches, provided that the co-ordinates agree in the areas where they overlap. The key insight is that there is no observer who can see all of space-time at once – any physical observer has to travel from one part of the manifold to another by a continuous route. Hence it does not matter if the observer cannot describe the entire manifold by a single map, so long as they have a series of maps that smoothly translate into one another as they travel along their chosen path – even if the maps used much later in the journey have no connection or overlap with the maps used early in the journey. Similarly, as we extend our measuring devices into new regimes, we must gradually replace them with new devices as we go. The eye is replaced with the microscope, the microscope with the electron microscope and the electron microscope with the particle accelerator, which now bears no resemblance to the eye, although they both gaze upon the same world.

Curiously, there was another man named Bridgman active around the same time, who is likely to be more familiar to artists: that is George Bridgman, author of Bridgman’s Complete Guide to Drawing From Life. Although they were two completely different Bridgmans, working in different disciplines, both of them were concerned with essentially the same problem: how to connect our internal conception of the world with the devices by which we measure the world. In the case of Percy Bridgman, it was a matter of connecting abstract physical quantities to their measurement devices, while George Bridgman aimed to connect the figure in the mind to the functions of the hands and eyes. We close with a quote from the artist:

“Indeed, it is very far from accurate to say that we see with our eyes. The eye is blind but for the idea behind the eye.”

[1] Everything I have written comes from Hasok Chang’s entry in the Stanford Encyclopedia of Philosophy on operationalism, which is both clearer and more thorough than my own ramblings.

[2] Readers interested in the finer points of Percy Bridgman’s work should see his Nobel prize lecture.